Fault-Tolerant Total Domination via Submodular Function Approximation

In total domination, given a graph $$G=(V,E)$$ , we seek a minimum-size set of nodes $$S\subseteq V$$ , such that every node in $$V \setminus S$$ has at least one neighbor in S and every node in S also has at least one neighbor in S. We define the fault-tolerant version of total domination, where we extend the requirement for nodes in $$V \setminus S$$ . Any node in $$V \setminus S$$ must have at least m neighbors in S. Let $$\varDelta $$ denote the maximum degree in G. We prove a first $$1 + \ln (\varDelta + m - 1)$$ approximation for fault-tolerant total domination. To prove our result, we develop a general submodular function approximation framework we believe is of independent interest.